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<b>let </b><font color="Maroon">a</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">b</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>;<br/>
<a NAME="E1:14"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:14"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">kk</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E3:14_1_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><b>set </b><font color="Maroon">T</font> = <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>;<br/><a NAME="E4:14_1_1_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:14_1_1_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="newton.html#T71">NEWTON:71</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/><a NAME="E3:14_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_1_1_2"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">T1</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span>;<br/><a NAME="E3:14_1_1_1_2_1_1_2"/>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E4:14_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><a NAME="E5:14_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">T2</font> = <font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span>;<br/><a NAME="E6:14_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><a NAME="E7:14_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i>, , , </i>;<br/><a NAME="E8:14_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><a NAME="E9:14_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kk</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E11:14_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:14_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">kk</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:14"/>
for <font color="Olive">c</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:14"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 ;<br/>


</div>
